We believe that these mixed structures naturally appear in a harmonic map compactification of the Hitchin and maximal components of the character variety for real Lie groups \(G\)  of rank 2. In this context, Labourie ([1]}), Collier ([2]}) and Collier-Tholozan-Toulisse ([3]}) proved that given a Hitchin or maximal representation \(\rho :\pi _{1}(S) \rightarrow G\)  there exists a unique \(\rho \) -equivariant minimal surface \(\widetilde{\Sigma }_{\rho }\)  in \(G/K\) , where \(K\)  is a maximal compact subgroup of \(G\) . One could then find a compactification of these components by studying the limiting behaviour of \(\widetilde{\Sigma }_{\rho _n}\)  when \(\rho _{n}\)  leaves all compact sets in the character variety. Up to subsequences, and after rescaling the metric on \(G/K\)  appropriately, \(\widetilde{\Sigma }_{\rho _n}\)  should converge to a subcomplex \(\widetilde{\Sigma }_{\infty }\subset B\) , where \(B\)  is a non-discrete Euclidean building modelled on the affine Weyl-group of \(G\) . We conjecture that \(\Sigma _{\infty }\)  is dual to a mixed structure as in Definition REF  where \(\overline{\mathfrak {a}^{+}}\)  is a Cartan subalgebra of the Lie algebra of \(G\)  and \(k\)  depends on the particular group. More precisely, we conjecture the following:
